Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936)
gp: K = bnfinit(y^21 - 45*y^19 - 30*y^18 + 729*y^17 + 972*y^16 - 4266*y^15 - 9180*y^14 - 6687*y^13 - 2872*y^12 + 131652*y^11 + 443208*y^10 + 46436*y^9 - 1786608*y^8 - 2821416*y^7 - 29552*y^6 + 4752864*y^5 + 6645312*y^4 + 4669568*y^3 + 1886976*y^2 + 419328*y + 39936, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936)
\( x^{21} - 45 x^{19} - 30 x^{18} + 729 x^{17} + 972 x^{16} - 4266 x^{15} - 9180 x^{14} - 6687 x^{13} + \cdots + 39936 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[19, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-460953958284049031411201347341619922421861679104\)
\(\medspace = -\,2^{15}\cdot 3^{23}\cdot 13^{2}\cdot 313^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(186.08\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(313\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}+\frac{1}{8}a^{10}-\frac{1}{4}a^{8}-\frac{3}{8}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{13}+\frac{1}{16}a^{12}-\frac{3}{32}a^{11}+\frac{1}{8}a^{10}+\frac{1}{16}a^{9}-\frac{3}{8}a^{8}+\frac{9}{32}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{128}a^{16}-\frac{1}{64}a^{15}-\frac{1}{128}a^{14}+\frac{1}{32}a^{13}-\frac{7}{128}a^{12}+\frac{5}{64}a^{11}-\frac{3}{64}a^{10}+\frac{1}{8}a^{9}-\frac{31}{128}a^{8}+\frac{19}{64}a^{7}-\frac{1}{2}a^{6}-\frac{3}{16}a^{5}-\frac{7}{32}a^{4}+\frac{1}{16}a^{3}+\frac{5}{16}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{512}a^{17}-\frac{5}{512}a^{15}-\frac{15}{256}a^{14}+\frac{1}{512}a^{13}+\frac{7}{128}a^{12}-\frac{25}{256}a^{11}+\frac{25}{128}a^{10}+\frac{1}{512}a^{9}+\frac{21}{64}a^{8}-\frac{29}{128}a^{7}+\frac{9}{64}a^{6}-\frac{51}{128}a^{5}+\frac{13}{32}a^{4}+\frac{7}{64}a^{3}-\frac{7}{32}a^{2}-\frac{5}{16}a-\frac{1}{8}$, $\frac{1}{4096}a^{18}-\frac{1}{2048}a^{17}+\frac{11}{4096}a^{16}+\frac{3}{1024}a^{15}-\frac{83}{4096}a^{14}-\frac{243}{2048}a^{13}-\frac{237}{2048}a^{12}-\frac{43}{512}a^{11}-\frac{679}{4096}a^{10}-\frac{365}{2048}a^{9}-\frac{237}{1024}a^{8}-\frac{115}{256}a^{7}-\frac{503}{1024}a^{6}-\frac{227}{512}a^{5}+\frac{27}{512}a^{4}+\frac{13}{128}a^{3}-\frac{5}{64}a^{2}-\frac{5}{16}a-\frac{11}{32}$, $\frac{1}{32768}a^{19}-\frac{1}{8192}a^{18}-\frac{17}{32768}a^{17}-\frac{5}{16384}a^{16}-\frac{459}{32768}a^{15}-\frac{11}{256}a^{14}+\frac{489}{16384}a^{13}-\frac{841}{8192}a^{12}-\frac{7095}{32768}a^{11}-\frac{1667}{8192}a^{10}+\frac{303}{1024}a^{9}-\frac{921}{4096}a^{8}-\frac{1343}{8192}a^{7}+\frac{61}{1024}a^{6}+\frac{1297}{4096}a^{5}+\frac{95}{2048}a^{4}-\frac{119}{256}a^{3}-\frac{73}{256}a^{2}+\frac{81}{256}a+\frac{51}{128}$, $\frac{1}{262144}a^{20}+\frac{1}{131072}a^{19}+\frac{23}{262144}a^{18}-\frac{15}{16384}a^{17}-\frac{839}{262144}a^{16}-\frac{673}{131072}a^{15}-\frac{1783}{131072}a^{14}+\frac{3081}{32768}a^{13}+\frac{6897}{262144}a^{12}-\frac{3883}{131072}a^{11}+\frac{2811}{32768}a^{10}-\frac{8705}{32768}a^{9}-\frac{31915}{65536}a^{8}-\frac{2889}{32768}a^{7}-\frac{4119}{32768}a^{6}-\frac{3175}{8192}a^{5}-\frac{2959}{8192}a^{4}-\frac{3}{2048}a^{3}+\frac{187}{2048}a^{2}+\frac{51}{512}a+\frac{169}{512}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{1423230507}{32768}a^{20}-\frac{987694371}{32768}a^{19}-\frac{63359822847}{32768}a^{18}+\frac{1273413177}{32768}a^{17}+\frac{1036646576013}{32768}a^{16}+\frac{663971180765}{32768}a^{15}-\frac{3266103895965}{16384}a^{14}-\frac{4266019310037}{16384}a^{13}-\frac{3596568165121}{32768}a^{12}-\frac{1591842952179}{32768}a^{11}+\frac{47118946374477}{8192}a^{10}+\frac{62498566834153}{4096}a^{9}-\frac{70219673623575}{8192}a^{8}-\frac{586950863244351}{8192}a^{7}-\frac{298279719711761}{4096}a^{6}+\frac{201724495125507}{4096}a^{5}+\frac{352775523393711}{2048}a^{4}+\frac{21644055695583}{128}a^{3}+\frac{10941039494541}{128}a^{2}+\frac{5797063604475}{256}a+\frac{319974128939}{128}$, $\frac{4398560037}{262144}a^{20}-\frac{1540346211}{131072}a^{19}-\frac{195776668397}{262144}a^{18}+\frac{161290035}{8192}a^{17}+\frac{3202898190813}{262144}a^{16}+\frac{1016081714835}{131072}a^{15}-\frac{10093477982067}{131072}a^{14}-\frac{3280022542839}{32768}a^{13}-\frac{11038780698347}{262144}a^{12}-\frac{2451671573247}{131072}a^{11}+\frac{72814117599855}{32768}a^{10}+\frac{192686524974459}{32768}a^{9}-\frac{218818726737687}{65536}a^{8}-\frac{905657504567829}{32768}a^{7}-\frac{917002555476307}{32768}a^{6}+\frac{156466028362041}{8192}a^{5}+\frac{543701556595773}{8192}a^{4}+\frac{133168398195873}{2048}a^{3}+\frac{67212537802839}{2048}a^{2}+\frac{4445181735579}{512}a+\frac{490029755125}{512}$, $\frac{32744277}{32768}a^{20}-\frac{11985237}{16384}a^{19}-\frac{1455944861}{32768}a^{18}+\frac{20874321}{8192}a^{17}+\frac{23809453701}{32768}a^{16}+\frac{7198853849}{16384}a^{15}-\frac{75113469675}{16384}a^{14}-\frac{5956834755}{1024}a^{13}-\frac{79418398859}{32768}a^{12}-\frac{17951630913}{16384}a^{11}+\frac{542141571123}{4096}a^{10}+\frac{1417190521719}{4096}a^{9}-\frac{1694782599759}{8192}a^{8}-\frac{6692314938087}{4096}a^{7}-\frac{6649038883067}{4096}a^{6}+\frac{296654805783}{256}a^{5}+\frac{3994743685347}{1024}a^{4}+\frac{968879429493}{256}a^{3}+\frac{485275006827}{256}a^{2}+\frac{15933771711}{32}a+\frac{3488902259}{64}$, $\frac{7181342523}{262144}a^{20}-\frac{2469763233}{131072}a^{19}-\frac{319764806611}{262144}a^{18}+\frac{563203773}{32768}a^{17}+\frac{5232183680979}{262144}a^{16}+\frac{1690679330649}{131072}a^{15}-\frac{16481380216941}{131072}a^{14}-\frac{5406487307211}{32768}a^{13}-\frac{18262990461845}{262144}a^{12}-\frac{4029031544229}{131072}a^{11}+\frac{118872725972301}{32768}a^{10}+\frac{316089763608293}{32768}a^{9}-\frac{351526750290297}{65536}a^{8}-\frac{14\!\cdots\!83}{32768}a^{7}-\frac{15\!\cdots\!41}{32768}a^{6}+\frac{253550330110773}{8192}a^{5}+\frac{892263692005287}{8192}a^{4}+\frac{219374649492119}{2048}a^{3}+\frac{111049188746769}{2048}a^{2}+\frac{7364261017971}{512}a+\frac{813949094111}{512}$, $\frac{691056225}{32768}a^{20}-\frac{59837175}{4096}a^{19}-\frac{30765810641}{32768}a^{18}+\frac{289865925}{16384}a^{17}+\frac{503373029133}{32768}a^{16}+\frac{80755595519}{8192}a^{15}-\frac{1585858881519}{16384}a^{14}-\frac{1036710880683}{8192}a^{13}-\frac{1749119629607}{32768}a^{12}-\frac{24168853791}{1024}a^{11}+\frac{2859831227133}{1024}a^{10}+\frac{30361129805283}{4096}a^{9}-\frac{34035962241159}{8192}a^{8}-\frac{71269268860227}{2048}a^{7}-\frac{144988638678391}{4096}a^{6}+\frac{48930258116205}{2048}a^{5}+\frac{42845467587693}{512}a^{4}+\frac{21039179323423}{256}a^{3}+\frac{10639131842781}{256}a^{2}+\frac{1409766219225}{128}a+\frac{38920366075}{32}$, $\frac{891016443}{131072}a^{20}-\frac{299010177}{65536}a^{19}-\frac{39696116595}{131072}a^{18}-\frac{10655523}{16384}a^{17}+\frac{649684067955}{131072}a^{16}+\frac{214981179449}{65536}a^{15}-\frac{2045440085085}{65536}a^{14}-\frac{679218476979}{16384}a^{13}-\frac{2303348395541}{131072}a^{12}-\frac{504321205509}{65536}a^{11}+\frac{14747762910897}{16384}a^{10}+\frac{39465143295613}{16384}a^{9}-\frac{42689639721177}{32768}a^{8}-\frac{184715798543823}{16384}a^{7}-\frac{190181181050717}{16384}a^{6}+\frac{31161881364813}{4096}a^{5}+\frac{111460350768159}{4096}a^{4}+\frac{27534152510719}{1024}a^{3}+\frac{13987951439553}{1024}a^{2}+\frac{930590366355}{256}a+\frac{103166317991}{256}$, $\frac{4793102095}{262144}a^{20}-\frac{1612332377}{131072}a^{19}-\frac{213528473447}{262144}a^{18}-\frac{1952521}{4096}a^{17}+\frac{3494617710327}{262144}a^{16}+\frac{1153765856265}{131072}a^{15}-\frac{11002839242233}{131072}a^{14}-\frac{3649413955725}{32768}a^{13}-\frac{12371819638177}{262144}a^{12}-\frac{2710565442093}{131072}a^{11}+\frac{79334200200269}{32768}a^{10}+\frac{212169520472049}{32768}a^{9}-\frac{230117573317989}{65536}a^{8}-\frac{993277031688879}{32768}a^{7}-\frac{10\!\cdots\!69}{32768}a^{6}+\frac{167780405678787}{8192}a^{5}+\frac{599195792660295}{8192}a^{4}+\frac{147954599726467}{2048}a^{3}+\frac{75139448282181}{2048}a^{2}+\frac{4997420787913}{512}a+\frac{553869135983}{512}$, $\frac{7017585281}{262144}a^{20}-\frac{2424721535}{131072}a^{19}-\frac{312440190441}{262144}a^{18}+\frac{336350945}{16384}a^{17}+\frac{5112101625017}{262144}a^{16}+\frac{1644208998879}{131072}a^{15}-\frac{16104733435703}{131072}a^{14}-\frac{5270419746583}{32768}a^{13}-\frac{17789863724943}{262144}a^{12}-\frac{3930440455595}{131072}a^{11}+\frac{116163916399115}{32768}a^{10}+\frac{308507153416223}{32768}a^{9}-\frac{344916038147243}{65536}a^{8}-\frac{14\!\cdots\!57}{32768}a^{7}-\frac{14\!\cdots\!07}{32768}a^{6}+\frac{248218852769657}{8192}a^{5}+\frac{870772019903313}{8192}a^{4}+\frac{213893291782781}{2048}a^{3}+\frac{108198841735611}{2048}a^{2}+\frac{7170738883411}{512}a+\frac{792089736713}{512}$, $\frac{13848523839}{32768}a^{20}-\frac{2400242457}{8192}a^{19}-\frac{616527035783}{32768}a^{18}+\frac{5986283601}{16384}a^{17}+\frac{10087259100275}{32768}a^{16}+\frac{808429252515}{4096}a^{15}-\frac{31780671578933}{16384}a^{14}-\frac{20765841410119}{8192}a^{13}-\frac{35020019937593}{32768}a^{12}-\frac{3873737160119}{8192}a^{11}+\frac{229241009122297}{4096}a^{10}+\frac{608292755517511}{4096}a^{9}-\frac{682661606261785}{8192}a^{8}-\frac{357010710778175}{512}a^{7}-\frac{29\!\cdots\!01}{4096}a^{6}+\frac{981040216557041}{2048}a^{5}+\frac{858401496031891}{512}a^{4}+\frac{421412777503847}{256}a^{3}+\frac{213055999974827}{256}a^{2}+\frac{28225573149481}{128}a+\frac{779069096339}{32}$, $\frac{2284753703}{262144}a^{20}-\frac{800246829}{131072}a^{19}-\frac{101692496479}{262144}a^{18}+\frac{336707571}{32768}a^{17}+\frac{1663687326015}{262144}a^{16}+\frac{527679592725}{131072}a^{15}-\frac{5242934175329}{131072}a^{14}-\frac{1703574752571}{32768}a^{13}-\frac{5732342746025}{262144}a^{12}-\frac{1273448058241}{131072}a^{11}+\frac{37822046086921}{32768}a^{10}+\frac{100082883176793}{32768}a^{9}-\frac{113685922587325}{65536}a^{8}-\frac{470417786473587}{32768}a^{7}-\frac{476260066966881}{32768}a^{6}+\frac{81285419568933}{8192}a^{5}+\frac{282401178964203}{8192}a^{4}+\frac{69163450166467}{2048}a^{3}+\frac{34905863535909}{2048}a^{2}+\frac{2308389636667}{512}a+\frac{254455240483}{512}$, $\frac{28477712647}{262144}a^{20}-\frac{9800803125}{131072}a^{19}-\frac{1268005108351}{262144}a^{18}+\frac{2306791853}{32768}a^{17}+\frac{20747554400959}{262144}a^{16}+\frac{6699751456301}{131072}a^{15}-\frac{65354527829953}{131072}a^{14}-\frac{21432066889615}{32768}a^{13}-\frac{72414096959881}{262144}a^{12}-\frac{15972045186633}{131072}a^{11}+\frac{471391930560273}{32768}a^{10}+\frac{12\!\cdots\!69}{32768}a^{9}-\frac{13\!\cdots\!41}{65536}a^{8}-\frac{58\!\cdots\!75}{32768}a^{7}-\frac{59\!\cdots\!45}{32768}a^{6}+\frac{10\!\cdots\!13}{8192}a^{5}+\frac{35\!\cdots\!59}{8192}a^{4}+\frac{869718730295795}{2048}a^{3}+\frac{440253774571733}{2048}a^{2}+\frac{29196046383399}{512}a+\frac{3227082902059}{512}$, $\frac{163828040121}{131072}a^{20}-\frac{57244584539}{65536}a^{19}-\frac{7292252098977}{131072}a^{18}+\frac{22657133535}{16384}a^{17}+\frac{119303955162017}{131072}a^{16}+\frac{37933398302275}{65536}a^{15}-\frac{375954352212911}{65536}a^{14}-\frac{122309934008817}{16384}a^{13}-\frac{411721774490743}{131072}a^{12}-\frac{91394275457607}{65536}a^{11}+\frac{27\!\cdots\!63}{16384}a^{10}+\frac{71\!\cdots\!19}{16384}a^{9}-\frac{81\!\cdots\!15}{32768}a^{8}-\frac{33\!\cdots\!53}{16384}a^{7}-\frac{34\!\cdots\!95}{16384}a^{6}+\frac{58\!\cdots\!39}{4096}a^{5}+\frac{20\!\cdots\!21}{4096}a^{4}+\frac{49\!\cdots\!97}{1024}a^{3}+\frac{25\!\cdots\!15}{1024}a^{2}+\frac{165832613294209}{256}a+\frac{18285576591437}{256}$, $\frac{1944355181}{262144}a^{20}-\frac{669404243}{131072}a^{19}-\frac{86574175285}{262144}a^{18}+\frac{80068021}{16384}a^{17}+\frac{1416554685893}{262144}a^{16}+\frac{457262860723}{131072}a^{15}-\frac{4462177954219}{131072}a^{14}-\frac{1463024718795}{32768}a^{13}-\frac{4942657833571}{262144}a^{12}-\frac{1090402922511}{131072}a^{11}+\frac{32184992237503}{32768}a^{10}+\frac{85557861889875}{32768}a^{9}-\frac{95253155150287}{65536}a^{8}-\frac{401432328442581}{32768}a^{7}-\frac{409316166581323}{32768}a^{6}+\frac{68665848411525}{8192}a^{5}+\frac{241509123885725}{8192}a^{4}+\frac{59370116104185}{2048}a^{3}+\frac{30051250314399}{2048}a^{2}+\frac{1992761747287}{512}a+\frac{220248713717}{512}$, $\frac{13932161361}{65536}a^{20}-\frac{2434824763}{16384}a^{19}-\frac{620139346817}{65536}a^{18}+\frac{7772505791}{32768}a^{17}+\frac{10145694486293}{65536}a^{16}+\frac{806210872407}{8192}a^{15}-\frac{31971759008167}{32768}a^{14}-\frac{20799357291053}{16384}a^{13}-\frac{35003425868487}{65536}a^{12}-\frac{3885720540973}{16384}a^{11}+\frac{3603637467035}{128}a^{10}+\frac{610631489787143}{8192}a^{9}-\frac{691999209424479}{16384}a^{8}-\frac{89673568550405}{256}a^{7}-\frac{29\!\cdots\!63}{8192}a^{6}+\frac{990569576823259}{4096}a^{5}+\frac{215385136584093}{256}a^{4}+\frac{422173869748159}{512}a^{3}+\frac{213129725965705}{512}a^{2}+\frac{28196818263151}{256}a+\frac{388614586573}{32}$, $\frac{108824112261}{262144}a^{20}-\frac{38052279855}{131072}a^{19}-\frac{4843862499949}{262144}a^{18}+\frac{15345187687}{32768}a^{17}+\frac{79246929163821}{262144}a^{16}+\frac{25178423857431}{131072}a^{15}-\frac{249730018335827}{131072}a^{14}-\frac{81214404144317}{32768}a^{13}-\frac{273337655327275}{262144}a^{12}-\frac{60693992724139}{131072}a^{11}+\frac{18\!\cdots\!83}{32768}a^{10}+\frac{47\!\cdots\!03}{32768}a^{9}-\frac{54\!\cdots\!83}{65536}a^{8}-\frac{22\!\cdots\!61}{32768}a^{7}-\frac{22\!\cdots\!47}{32768}a^{6}+\frac{38\!\cdots\!35}{8192}a^{5}+\frac{13\!\cdots\!53}{8192}a^{4}+\frac{32\!\cdots\!05}{2048}a^{3}+\frac{16\!\cdots\!23}{2048}a^{2}+\frac{110089154956869}{512}a+\frac{12137624181617}{512}$, $\frac{68675232643}{262144}a^{20}-\frac{24002666961}{131072}a^{19}-\frac{3056829307547}{262144}a^{18}+\frac{9565925579}{32768}a^{17}+\frac{50010772422203}{262144}a^{16}+\frac{15896898600505}{131072}a^{15}-\frac{157596692331109}{131072}a^{14}-\frac{51264028138655}{32768}a^{13}-\frac{172552510404621}{262144}a^{12}-\frac{38308260096245}{131072}a^{11}+\frac{11\!\cdots\!85}{32768}a^{10}+\frac{30\!\cdots\!17}{32768}a^{9}-\frac{34\!\cdots\!49}{65536}a^{8}-\frac{14\!\cdots\!39}{32768}a^{7}-\frac{14\!\cdots\!05}{32768}a^{6}+\frac{24\!\cdots\!09}{8192}a^{5}+\frac{84\!\cdots\!95}{8192}a^{4}+\frac{20\!\cdots\!11}{2048}a^{3}+\frac{10\!\cdots\!05}{2048}a^{2}+\frac{69499416229583}{512}a+\frac{7663016252095}{512}$, $\frac{16131595039}{262144}a^{20}-\frac{6166547377}{131072}a^{19}-\frac{716493741751}{262144}a^{18}+\frac{3989675241}{16384}a^{17}+\frac{11711172216487}{262144}a^{16}+\frac{3363165461361}{131072}a^{15}-\frac{36980283506537}{131072}a^{14}-\frac{11442861720721}{32768}a^{13}-\frac{37880137651537}{262144}a^{12}-\frac{8683388532357}{131072}a^{11}+\frac{267129343724853}{32768}a^{10}+\frac{689478079991521}{32768}a^{9}-\frac{867015141789045}{65536}a^{8}-\frac{32\!\cdots\!75}{32768}a^{7}-\frac{31\!\cdots\!89}{32768}a^{6}+\frac{594525019097759}{8192}a^{5}+\frac{19\!\cdots\!03}{8192}a^{4}+\frac{466405922965411}{2048}a^{3}+\frac{231893707338117}{2048}a^{2}+\frac{15126936126117}{512}a+\frac{1645600889095}{512}$, $\frac{206442231459}{262144}a^{20}-\frac{71530154517}{131072}a^{19}-\frac{9190764080283}{262144}a^{18}+\frac{5492186957}{8192}a^{17}+\frac{150374663576875}{262144}a^{16}+\frac{48227586575205}{131072}a^{15}-\frac{473762573583557}{131072}a^{14}-\frac{154815449424193}{32768}a^{13}-\frac{522200243316941}{262144}a^{12}-\frac{115512343041753}{131072}a^{11}+\frac{34\!\cdots\!29}{32768}a^{10}+\frac{90\!\cdots\!93}{32768}a^{9}-\frac{10\!\cdots\!25}{65536}a^{8}-\frac{42\!\cdots\!63}{32768}a^{7}-\frac{43\!\cdots\!97}{32768}a^{6}+\frac{73\!\cdots\!23}{8192}a^{5}+\frac{25\!\cdots\!91}{8192}a^{4}+\frac{62\!\cdots\!51}{2048}a^{3}+\frac{31\!\cdots\!77}{2048}a^{2}+\frac{210451046194333}{512}a+\frac{23236325997107}{512}$, $\frac{168986791913}{262144}a^{20}-\frac{58495299211}{131072}a^{19}-\frac{7523398874865}{262144}a^{18}+\frac{17359009407}{32768}a^{17}+\frac{123094645622513}{262144}a^{16}+\frac{39518218798707}{131072}a^{15}-\frac{387802811482895}{131072}a^{14}-\frac{126792892486137}{32768}a^{13}-\frac{427839658989063}{262144}a^{12}-\frac{94584009176631}{131072}a^{11}+\frac{27\!\cdots\!03}{32768}a^{10}+\frac{74\!\cdots\!15}{32768}a^{9}-\frac{83\!\cdots\!15}{65536}a^{8}-\frac{34\!\cdots\!85}{32768}a^{7}-\frac{35\!\cdots\!15}{32768}a^{6}+\frac{59\!\cdots\!51}{8192}a^{5}+\frac{20\!\cdots\!97}{8192}a^{4}+\frac{51\!\cdots\!61}{2048}a^{3}+\frac{26\!\cdots\!07}{2048}a^{2}+\frac{172428678676145}{512}a+\frac{19042079271253}{512}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 681712691863000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{19}\cdot(2\pi)^{1}\cdot 681712691863000000 \cdot 1}{2\cdot\sqrt{460953958284049031411201347341619922421861679104}}\cr\approx \mathstrut & 1.65383582609970 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 45*x^19 - 30*x^18 + 729*x^17 + 972*x^16 - 4266*x^15 - 9180*x^14 - 6687*x^13 - 2872*x^12 + 131652*x^11 + 443208*x^10 + 46436*x^9 - 1786608*x^8 - 2821416*x^7 - 29552*x^6 + 4752864*x^5 + 6645312*x^4 + 4669568*x^3 + 1886976*x^2 + 419328*x + 39936);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$C_3^7.C_2\wr C_7:C_3$ (as 21T137):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 5878656 |
| The 183 conjugacy class representatives for $C_3^7.C_2\wr C_7:C_3$ |
| Character table for $C_3^7.C_2\wr C_7:C_3$ |
Intermediate fields
| 7.7.9597924961.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 42 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $21$ | $21$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{5}$ | $21$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
| 2.6.6.2 | $x^{6} + 6 x^{5} + 14 x^{4} + 24 x^{3} + 44 x^{2} + 8 x + 72$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2]^{6}$ | |
|
\(3\)
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
| 3.9.9.7 | $x^{9} - 6 x^{7} + 9 x^{6} - 36 x^{5} - 36 x^{4} + 459 x^{3} - 108 x^{2} - 54 x + 27$ | $3$ | $3$ | $9$ | $C_3^2 : S_3 $ | $[3/2, 3/2]_{2}^{3}$ | |
| 3.9.9.4 | $x^{9} - 24 x^{8} + 318 x^{7} - 189 x^{6} + 1080 x^{5} + 2826 x^{4} + 1350 x^{3} - 108 x^{2} - 54 x + 27$ | $3$ | $3$ | $9$ | $(C_3^3:C_3):C_2$ | $[3/2, 3/2, 3/2]_{2}^{3}$ | |
|
\(13\)
| 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(313\)
| $\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $6$ | $3$ | $2$ | $4$ | ||||
| Deg $6$ | $3$ | $2$ | $4$ |